Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,16)(13,15);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,18,23,26)(19,28,31,32)(20,25,24,30)(21,27,22,29);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,19,23,31)(18,24,26,20)(21,25,22,30)(27,32,29,28);(9,12,13,11)(10,16,15,14)(17,20,23,24)(18,25,26,30)(19,29,31,27)(21,32,22,28);(1,4,5,3)(2,6,8,7)(17,21)(18,26)(19,30)(22,23)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498098);